Given the polynomial $p(x) = x ^ 3 + ax ^ 2 + 5$, I have to find the value of $a$ for which the polynomial is factorizable as the product of three first degree factors in $Z /_{3Z}$.
The polynomial is reducible for $a = 0 (mod\;3)$ and for $a = 2 (mod\;3)$
$a = 0 \implies x^3+2 = (x-1)(x^2+2x+1)$
for $a = 1$ there is no factoring
Therefore the polynomial cannot be factored as a product of prime factors for any value of $a$.What am I doing wrong?
Notice that $$ x^3+2=x^3-1=(x-1)(x^2+x+1)=(x-1)(x^2-2x+1)=(x-1)^3 (mod \ 3) $$ For $a=2 (mod \ 3)$ $$ x^3+2x^2+2=(x-2)(x^2+4x-1)=(x-2)(x^2+x-1)(mod \ 3) $$ But it is not possible to factorize $(x^2+x-1)$ in $\mathbb{Z}_3$